On Cohen and Prikry Forcing Notions

TL;DR

This paper demonstrates adding Cohen subsets to a cardinal via Prikry forcing, introduces a strengthened non-Galvin property, and explores extender-based Prikry forcings, answering open questions in set theory.

Contribution

It shows the possibility of adding Cohen subsets with Prikry forcing over a measurable cardinal and introduces a strengthened non-Galvin property, improving previous results.

Findings

Adding ppa^+-Cohen subsets to ppa with Prikry forcing is consistent.

A strengthened non-Galvin property is introduced and shown consistent with a single measurable cardinal.

Examines extender-based Prikry forcings in relation to a question by H. Woodin.

Abstract

We show that it is possible to add $\kappa^+-$Cohen subsets to $\kappa$ with a Prikry forcing over $\kappa$. This answers a question from \cite{HayutBenhanouGitik}. A strengthening of non-Galvin property is introduced. It is shown to be consistent using a single measurable cardinal which improves a previous result by S. Garti, S. Shelah, and the first author \cite{BenhamouGartieShelah}. A situation with extender-based Prikry forcings is examined. This relates to a question of H. Woodin.